Topologies on Types. Drew Fudenberg Harvard University. First Draft: April 2004 This Draft: August Abstract

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1 Topologies on Types Eddie Dekel Northwestern University and Tel Aviv University Drew Fudenberg Harvard University Stephen Morris Princeton University First Draft: April 2004 This Draft: August 2005 Abstract We de ne and analyze "strategic topologies" on types, under which two types are close if their strategic behavior will be similar in all strategic situations. To operationalize this idea, we adopt interim rationalizability as our solution concept, and de ne a metric topology on types in the Harsanyi-Mertens-Zamir universal type space. This topology is the coarsest metric topology generating upper and lower hemicontinuity of rationalizable outcomes. While upper strategic convergence is equivalent to convergence in the product topology, lower strategic convergence is a strictly stronger requirement, as shown by the electronic mail game. Nonetheless, we show that the set of " nite types" (types describable by nite type spaces) are dense in the lower strategic topology. JEL classi cation and keywords: C70, C72, rationalizability, incomplete information, common knowledge, universal type space, strategic topology. Previous versions were circulated under the title "Dense Types". We are grateful for comments from seminar participants at Stanford University, the University of Essex, the Cowles Foundation conference on Uncertainty in Economic Theory and the Third International PIER/IGIER Conference on Economic Theory. We are grateful to Dirk Bergemann, Je Ely, Yossi Feinberg, Aviad Heifetz, Atsushi Kajii, Bart Lipman, Aldo Rustichini, Muhamet Yildiz and Shmuel Zamir for helpful discussions; Marcin Peski for detailed comments; and Satoru Takahashi for expert proof reading. This material is based upon work supported by the National Science Foundation under Grants No and

2 1 Introduction The universal type space proposed by Harsanyi (1967/68) and constructed by Mertens and Zamir (1985) shows how (under some technical assumptions) any incomplete information about a strategic situation can be embedded in a single "universal" type space. As a practical matter, applied researchers do not work with that type space but with smaller subsets of the universal type space. Mertens and Zamir showed that nite type spaces are dense under the product topology, but under this topology the rationalizable actions of a given type may be very di erent from the rationalizable actions of a sequence of types that approximate it. 1 This leads to the question of whether and how one can use smaller type spaces to approximate the predictions that would be obtained from the universal type space. To address this question, we de ne and analyze "strategic topologies" on types, under which two types are close if their strategic behavior is similar in all strategic situations. There are three ingredients that need to be formalized in this approach: how we vary the "strategic situations", what "strategic behavior" do we consider (i.e., what solution concept), and what we mean by "similar." To de ne "strategic situations," we start with a given space of uncertainty,, and a type space over that space. Holding these xed, we then consider all possible nite action games where payo s depend on the actions chosen by the players and the state of Nature. Thus we vary the game while holding the type space xed. This is at odds with a broader interpretation of the universal type space which describes all possible uncertainty including payo functions and actions: see the discussion in Mertens, Sorin and Zamir (1994, Remark 4.20b). According to this latter view one cannot identify "higher order beliefs" independent of payo s in the game. In contrast, our de nition of a strategic topology relies crucially on making this distinction. We start with higher-order beliefs about abstract payo -relevant states, then allow payo s to vary by changing how payo s depend on. We are thus implicitly assuming that any "payo relevant state" can be associated with any payo s and actions. This is analogous to Savage s assumption that all acts are possible, and thus implicitly that any "outcome" is consistent with any payo -relevant state. This separation is natural for studying the performance of di erent mechanisms under various informational 1 This is closely related to the di erence between common knowledge and mutual knowledge of order n that is emphasized by Geanakoplos and Polemarchakis (1982) and Rubinstein (1989). 2

3 assumptions. 2 Our notion of "strategic behavior" is the set of (correlated) interim rationalizable actions that we analyzed in Dekel, Fudenberg and Morris (2005). The rationalizable actions are those obtained by the iterative deletion of all actions that are not best responses given a type s beliefs over others types and Nature and any (perhaps correlated) beliefs about which actions are played at a given type pro le and payo -relevant state. Under correlated interim rationalizability, a player s beliefs allow for arbitrary correlation between other players actions and the payo state; in the complete information case, this de nition reduces to the standard de nition of correlated rationalizability (e.g., as in Brandenburger and Dekel (1987)). One might be interested in using equilibrium behavior, as opposed to rationalizability, as the benchmark for rational play. However, the restriction to equilibrium only has bite when additional structure, such as a common prior, is imposed. This is because the set of interim rationalizable actions for a given type t i is equal to the set of actions that might be played in some equilibrium in a larger type space by a type that has the same beliefs and higher order beliefs about Nature as type t i. 3 Thus, while one might be interested in using equilibrium behavior, as opposed to rationalizability, as the benchmark for rational play, the restriction to equilibrium only has bite when additional structure, such as a common prior, is imposed. Moreover, the set of interim rationalizable actions depends only on types beliefs and higher-order beliefs about Nature. 4 It remains to explain our notion of "similar" behavior. Our goal is to nd a topology on types that is ne enough that the set of correlated rationalizable actions has the continuity properties that the best response correspondences, rationalizable actions, and Nash equilibria 2 Wilson (1987) argued that an important task of mechanism design is to identify mechanisms that perform well under a variety of informational assumptions, and a number of recent works have pursued this agenda, see, e.g., Neeman (2004) and Bergemann and Morris (2004). 3 See Dekel, Fudenberg and Morris (2005); this is analogous to the property identi ed by Brandenburger and Dekel (1987) in the context of complete information. 4 As noted by Dekel, Fudenberg and Morris (2005) and Ely and Peski (2005), the set of independent interim rationalizable actions can vary across types with the same beliefs and higher order beliefs about Nature. This is because redundant types may di er in the extent they believe that others actions are correlated with payo relevant states. 3

4 all have in complete-information games, while still being coarse enough to be useful. 5 A review of those properties helps clarify our work. Fix a family of complete information games with payo functions u that depend continuously on a parameter, where lies in a metric space. Because best responses include the case of exact indi erence, the set of best responses for player i to a xed opponents strategy pro le a j, denoted BR i (a j ; ), is upper hemi-continuous but not lower hemi-continuous in, i.e. it may be that n! ; and a i 2 BR i (a j ; ); but there is no sequence a n i 2 BR i (a j ; n ) that converges to a i. However, the set of "-best responses, 6 BR i (a j ; "; ); is well behaved: if n! ; and a i 2 BR i (a j ; "; ); then for any a n i! a i there is a sequence " n! 0 such that a n i 2 BR i (a j ; " + " n ; n ). In particular, the smallest " for which a i 2 BR i (a j ; "; ) is a continuous function of. Moreover the same is true for the set of all "-Nash equilibria (Fudenberg and Levine (1986)) and for the set of "-rationalizable actions. 7 That is, the " that measures the departure from best response or equilibrium is continuous. Our goal is to nd the coarsest topology on types such that the same property holds. Thus for a xed game and action, we identify for each type of a player, the smallest " for which the action is rationalizable. The distance between a pair of types (for a xed game and action) is the di erence between those smallest ". Our strategic topology requires that, for any convergent sequence, this distance tends to zero pointwise for any action and game. We identify a metric for this topology. The strategic topology requires both a lower hemi-continuity property (the smallest " does not jump down in limit) and an upper hemicontinuity property (the smallest " does not jump up in the limit). We show that the upper convergence is equivalent to convergence in the product topology (theorem 2) and that lower convergence implies product convergence, and thus upper convergence (theorem 1). Our main result is that nite types are dense in the strategic topology (theorem 3). Thus nite type spaces do approximate the universal type space, so that the strategic behavior of 5 Topology P is ner than topology P 0 if every open set in P is contained in an open set in P 0. The use of a very ne topology such as the discrete topology makes continuity trivial, but it also makes it impossible to approximate one type with another; hence our search for a relatively coarse topology. We will see that our topology is the coarsest mertrizable topology with the desired continuity property; this leaves open whether there are other, non-metrizable, topologies that have this property 6 An action is an "-best response if it gives a payo within " of the best response. 7 This can be shown by using the equivalence between rationalizable actions and a posteriori equilibria (Brandenburger and Dekel (1987)) and then applying the result for Nash equilibrium. 4

5 any type can be approximated by a nite type. However, this does not imply that the set of nite types is large. In fact, while nite types are dense in the strategic topology (and the product topology), they are small in the sense of being category 1 in the product topology and the strategic topology. Our paper follows Monderer and Samet (1996) and Kajii and Morris (1997) in seeking to characterize "strategic topologies" that yield lower and upper hemi-continuity of strategic outcomes. The earlier papers de ned topologies on common prior information systems with a countable number of types, and common p-belief was central to the characterizations. 8 This paper performs an analogous exercise on the set of all types in the universal type space without the common prior assumption. We do not have a characterization of this strategic topology in terms of beliefs, so we are unable to pin down the relation to these earlier papers. The paper is organized as follows. Section 2 reviews the electronic mail game and the failure of lower hemicontinuity (but not upper hemicontinuity) of rationalizable outcomes with respect to the product topology. The universal type space is described in section 3 and the incomplete information games and interim rationalizable outcomes we will analyze are introduced in section 4. The strategic topology is de ned in section 5 and our main results about the strategic topology are reported in section 6. The concluding section, 7, contains some discussion of the interpretation of our results, the "genericity" of nite types and an alternative stronger uniform strategic topology on types. 2 Electronic Mail Game To introduce the basic issues we use a variant of Rubinstein s (1989) electronic mail game that illustrates the failure of lower hemi-continuity in the product topology (de ned formally below). Speci cally, we will use it to provide a sequence of types, t ik, that converge to a type t i1 in the product topology, while there is an action that is rationalizable for t i1 but is not " k -rationalizable for t k for any sequence " k! 0. Thus the set "blows up" at the limit, and the lower-hemicontinuity property discussed in the introduction is not satis ed. Intuitively, for rationalizable play, the tails of higher order beliefs matter, but the product topology is 8 For Monderer and Samet (1986), an information system was a collection of partitions on a xed state space with a given prior. space. For Kajii and Morris (1997), an information system was a prior on a xed type 5

6 insensitive to the tails. On the other hand the set of rationalizable actions does satisfy an upper hemi-continuity property with respect to the sequence of types t 1k converging in the product topology to t 11 : x any sequence " k! 0 and suppose some action is " k -rationalizable for type t 1k for all k; then that action is 0-rationalizable for type t 11. In section 6 we show that product convergence is equivalent to this upper hemi-continuity property in general. Example: Each player has two possible actions A 1 = A 2 = fn; Ig ("not invest" or "invest"). There are two payo states, = f0; 1g. In payo state 0, payo s are given by the following matrix: = 0 N I N 0; 0 0; 2 I 2; 0 2; 2 In payo state 1, payo s are given by: = 1 N I N 0; 0 0; 2 I 2; 0 1; 1 Player i s types are T i = ft i1 ; t i2 ; ::::g[ft i1 g. Beliefs are generated by the following common prior on the type space: = 0 t 21 t 22 t 23 t 24 t 21 t 11 (1 ) t t t t

7 = 1 t 21 t 22 t 23 t 24 t 21 t t 12 (1 ) (1 ) (1 ) (1 ) t 13 0 (1 ) (1 ) 3 (1 ) (1 ) t (1 ) (1 ) 5 (1 ) (1 ) t where, 2 (0; 1). There is an intuitive sense in which the sequence (t 1k ) 1 k=1 converges to t 11. Observe that type t 12 of player 1 knows that = 1 (but does not know if player 2 knows it). Type t 13 of player 1 knows that = 1, knows that player 2 knows it (and knows that 1 knows it), but does not know if 2 knows that 1 knows that 2 knows it. For k 3, each type t 1k knows that = 1, knows that player 2 knows that 1 knows... (k 2 times) that = 1. But for type t 11, there is common knowledge that = 1. Thus type t 1k agrees with type t 11 up to 2k 3 levels of beliefs. We will later de ne more generally the idea of product convergence of types, i.e., the requirement that kth level beliefs converge for every k. In this example, (t 1k ) 1 k=1 converges to t 11 in the product topology. We are interested in the "-rationalizable actions in this game. We will provide a formal de nition shortly, but the idea is that we will iteratively delete an action for a type at round k if that action is not an "-best response for any belief over the action-type pairs of the opponent that survived to round k 1. Clearly, both N and I are 0-rationalizable for types t 11 and t 21 of players 1 and 2, respectively. But action N is the unique "-rationalizable action for all types of each player i except t i1, for every " < (note that > 1 ). Clearly, I is not "-rationalizable for type t 11, since the expected payo from action N is 0 independent of player 2 s action, whereas the payo from action I is 2. Now suppose we can establish that I is not "-rationalizable for types t 11 through t 1k. Type t 2k s expected payo from action I is at most 1 2 (1) ( 2) = < 1 2. Thus I is not "-rationalizable for type t 2k. A symmetric argument establishes if I is not "-rationalizable for types t 21 through t 2k, then I is not "-rationalizable for type t 1;k+1. Thus the conclusion holds by induction. We conclude that strategic outcomes are not continuous in the product topology. Thus 7

8 the denseness of nite types in the product topology does not imply that they will be dense in our strategic topology. However, this version of the game is not a counterexample to the denseness of nite types, since the limit type t 11 is itself a nite type. 3 Types In games of incomplete information, a player s chosen action can depend not only on the player s payo function, but also on his beliefs about the opponents payo s, his beliefs about the opponents beliefs, and so on: in short, on the in nite hierarchy of beliefs. To model this, we look at type spaces that are subsets of the "universal type space" constructed by Mertens and Zamir (1985). 9 There are two agents, 1 and 2; we denote them by i and j = 3 i. 10 Let be a nite set representing possible payo -relevant moves by Nature. 11 Throughout the paper, we write (S) for the set of probability measures on the Borel eld of any topological space S. Let X 0 = X 1 = X 0 (X 0 ). X k = X k 1 (X k 1 ). where (X k ) is endowed with the topology of weak convergence of measures (i.e. the "weak" topology) and each X k is given the product topology over its two components. A type t is a hierarchy of beliefs t = ( 1 ; 2 ; :::) 2 1 k=0 (X k). We are interested in the set of belief 9 See also Brandenburger and Dekel (1993), Heifetz (1993), and Mertens, Sorin and Zamir (1994). Since is nite, the construction here yields the same universal space with the same - elds as the topology-free construction of Heifetz and Samet (1998); see Dekel, Fudenberg and Morris (2005, Lemma 1). Since we are interested in nding a topology, the Heifetz and Samet topology-free approach is more suitable than assuming a topology for this construction. But since the measure-theoretic properties of the two approaches are equivalent we present our work using the more familiar topological constructions. 10 We restrict the analysis to the two player case for notational convenience. We do not think that there would be any di culties extending the results to any nite number of players. 11 We choose to focus on nite as here the choice of a topology is obvious, while in larger spaces the topology on types will depend on the underlying topology on. 8

9 hierarchies in which it is common knowledge that the marginal of beliefs k on any level X k 0, for k 0 k 1 is equal to k We write T for the set of all such types and denote by k (t) the kth coordinate of t = ( 1 ; 2 ; :::). Mertens and Zamir (1985) characterize and prove the existence of induced beliefs for each type over T, denoted : T! (T ); the universal type space is (T i ; i ) 2 i=1. We will focus on types which give rise to distinct hierarchies of beliefs, and ignore the possibility of what Mertens and Zamir (1985) labelled "redundant types" i.e., types who agree about their higher order beliefs and disagree only in their beliefs over others redundant types. Previous work (Bergemann and Morris (2001), Battigalli and Siniscalchi (2003), Dekel, Fudenberg, and Morris (2005), Ely and Peski (2005)) has emphasized that redundant types matter for certain solution concepts (such as Nash equilibrium, correlated equilibrium and independent interim rationalizability). In contrast, Dekel, Fudenberg, and Morris (2005) show that two types with the same hierarchy of beliefs, i.e., that map to the same point in the universal type space, have the same set of (correlated) interim rationalizable actions. This is the reason that we use (correlated) interim rationalizability as the solution concept. Our main result will concern " nite types," meaning types in the universal type space that belong to nite belief-closed subset of the universal type space. To de ne them more precisely, we need to de ne nite type spaces and discuss how they can be embedded in the universal type space. De nition 1 A nite type space is any collection T = (T i ; i ) 2 i=1, where each T i is nite and each i : T i! (T j ). For any nite belief-closed type space (T i ; i ) 2 i=1 ; and any t i 2 T i, we can calculate the in nite hierarchy of i s beliefs about, beliefs about beliefs, etc. Thus each type t i can be identi ed with a unique type t 2 T. Formally, for each k = 1; 2; :::, we can de ne kth level beliefs iteratively as follows. Let be de ned as: b i1 : T i! () b i1 [t i ] () = X Thus b i1 : T i! (X 0 ). Now inductively let t j 2T j b i [t i ] (t j ; ) ; b ik : T i! (X k 1 ) 9

10 be de ned as: X b ik [t i ] ( j;k 1 ; ) = b i [t i ] (t j ; ) ; ft j 2T j :b j;k 1 (t j )= j;k 1g Thus b ik : T i! (X k 1 ). Let b i (t i ) = (b ik (t i )) 1 k=1, so b i : T i! T. We say that a type t 2 T is " nite" if there exists a nite type space T containing a type t i with the same higher-order beliefs as t (i.e., b i (t i ) = t). That is, a type is nite if it belongs to a nite belief-closed subset of the universal type space. The most commonly used topology on the universal type space is the "product topology" on the hierarchy: De nition 2 t n i! t i if, for each k, ik (t n i )! ik (t i ) as n! 1. Here, the convergence of beliefs at a xed level in the hierarchy, represented by!, is with respect to the topology of weak convergence of measures. This notion of convergence generates a topology that has been the most used in the study of the universal type space. See, for example, Lipman (2003) and Weinstein and Yildiz (2003). However, our purpose is to construct a topology that re ects strategic behavior, and as the electronic mail game suggests, the product topology is too coarse to have the continuity properties we are looking for. 4 Games and Interim Rationalizability Our interest is in the implications of these types for play in the family of games G de ned as follows. A game G consists of, for each player i, a nite set of possible actions A i and a payo function g i, where g i : A! [ M; M] and M is an exogenous bound on the scale of the payo s. Here we restate de nitions and results from our companion paper, Dekel, Fudenberg and Morris (2005). In that paper, we varied the type space and held xed the game G being played and the """ in the de nition of "-best response. In this paper, we x the type space to be the universal type space (and nite belief-closed subsets of it), but we vary the game G and parameter ", so we make the dependence of the solution on G and " explicit. For any subset of actions for all types, we rst de ne the best replies when beliefs over opponents strategies are restricted to those actions. For any measurable strategy pro le 10

11 of the opponents, j : T j! (A j ), where throughout j 6= i, and any belief over opponents types and the state of Nature, i (t i ) 2 T j, denote the induced belief over the space of types, Nature and actions by ( i (t i ) ; j ) 2 T j A j, where for measurable F T j ; ( i (t i ) ; j ) (F f; a j g) = R F j (t j ; ) [a j ] i (t i ) [dt j ; ]. De nition 3 The correspondence of best replies for all types given a subset of actions for all types is denoted BR : 2 A i T i! 2 A i T i and is de ned as follows. First, i2i given a speci cation of a subset of actions for each possible type of opponent, denoted by E j = E tj, with E tj A j for all t j and j 6= i, we de ne the " best replies for t i t j 2T j j6=i in game G as Tj A j such that >< (i) [f(t j ; ; a j ) : a j 2 E j g] = 1 BR i (t i ; E j ) = a i 2 A i (ii) marg T = i (t i ) " # R gi (a i ; a j ; ) >: (iii) d g i (a 0 i; a j ; ) (t j ;;a j ) i2i " for all a 0 i 2 A i Remark 1 In cases where E j is not measurable, we interpret [f(t j ; ; a j ) : a j 2 E j g] = 1 as saying that there is a measurable subset E 0 E j such that v [ E 0 ] = 1: The solution concept and closely related notions with which we work in this paper are given below. De nition 4 1. The interim rationalizable set, R = the largest xed point of BR. 2. R 0 i A T i i ; R k BR (R k 1 ), and R 1 \ 1 k=1 R k. (R i (t i )) ti 2Ti i2i A T i 9 >= >; i2i, is 3. Let S = (S 1 ; S 2 ), where each S i : T! 2 A i?; S is a best-reply set if for each ti and a i 2 S i (t i ), there exists a measurable 2 T j A j such that (i) [f(t j ; ; a j ) : a j 2 S j g] = 1 (ii) marg T = i (t i ) " # R gi (a i ; a j ; ) (ii) d g i (a 0 i; a j ; ) (t j ;;a j ) " for all a 0 i 2 A i 11

12 Arguments in Dekel, Fudenberg and Morris (2005) establish that these sets are wellde ned, and the relationships among them. These are restated in the following results that are taken from that work. Result 1 1. R i;k (t i ) = R i;k (t 0 i) if k (t i ) = k (t 0 i). That is, types with the same k th -order beliefs have the same k th -order rationalizable sets. 2. If Sc T for all c in some index set C are best-reply sets then [ c Sc T is a best-reply set. 3. The union of all best-reply sets is a best reply set. It is also the largest xed point of BR. 4. R equals R R i;k and R i;1 are measurable functions from T i! 2 A i =;, and for each action a i and each k the sets ft : a i 2 R i;k (t)g and ft : a i 2 R i;1 (t)g are closed. In de ning our strategic topology, we will exploit the following closure properties of "-rationalizable sets as a function of ": Lemma 1 For each k = 0; 1; :::, if " n # " and a i 2 Ri k (t i ; G; " n ) for all n, then a i 2 Ri k (t i ; G; "). Proof. We will prove this by induction. It is vacuously true for k = 0 Suppose that it holds true up to k 8 k i (t i ; ) = >< >: 2 (A j ) : 1. Let (a j ; ) = R [dt Tj j ; ; a j ] for some 2 Tj A j such that (t j ; ; a j ) : a j 2 R k 1 j (t j ; G; ") = 1 and marg T = i (t i ) 9 >=. >; The sequence k i (t i ; " n ) is decreasing (under set inclusion) and converges (by -additivity) to k i (t i ; "); moreover, in Dekel, Fudenberg and Morris (2005) we show that each k i (t i ; " n ) is compact. Let 8 < k i (t i ; a i ; ) = : 2 (A j ) : X a j ; (a j ; ) 12 " gi (a i ; a j ; ) g i (a 0 i; a j ; ) # 9 = for all a 0 i 2 A i ;.

13 The sequence k i (t i ; a i ; " n ) is decreasing (under set inclusion) and converges to k i (t i ; a i ; "). Now a i 2 R k i (t i ; G; " n ) for all n ) ) k i (t i ; " n ) \ k i (t i ; a i ; " n ) 6=? for all n k i (t i ; ") \ k i (t i ; a i ; ") 6=? ) a i 2 R k i (t i ; G; "), where the second implication follows from the nite intersection property of compact sets. Proposition 1 If " n # " and a i 2 R i (t i ; G; " n ) for all n, then a i 2 R i (t i ; G; "). Proof. a i 2 R i (t i ; G; " n ) for all n ) a i 2 R k i (t i ; G; " n ) for all n and k ) a i 2 R k i (t i ; G; ") for all k, by the above lemma ) a i 2 R i (t i ; G; "). Corollary 1 For any t i, a i and G exists. min f" : a i 2 R i (t i ; G; ")g 5 The Strategic Topology Our goal is to nd the coarsest topology on types so that the "-best-response correspondence and "-rationalizable sets have continuity properties such as those satis ed by "-Nash equilibrium and "-rationalizability in complete information games with respect to the payo s. For any xed game and action, we de ne the distance between a pair of types as the di erence between the smallest " that would make that action "-rationalizable in that game. Thus h i (t i ja i ; G) = minf" : a i 2 R i (t i ; G; ")g d (t i ; t 0 ija i ; G) = jh i (t i ja i ; G) h i (t 0 ija i ; G)j We will write G m for the collection of games where each player has at most m actions. 13

14 De nition 5 (Upper Strategic Convergence) ((t n i ) 1 n=1 ; t i) satisfy the upper strategic convergence property (written t n i all n; a i ; G 2 G m.! U t i ) if 8m, 9 " n! 0 s.t. h i (t i ja i ; G) < h i (t n i ja i ; G) + " n for This implies that if " n! 0 and a i 2 R i (t n i ; G; " n ) for each n, then a i 2 R i (t i ; G; 0). 12 We do not require convergence uniformly over all games, since an upper bound on the number of actions m is xed before the approximating sequence " n is chosen. Requiring uniformity over all games would considerably strengthen the topology, as brie y discussed in Section 7.2. On the other hand, it will be clear from our arguments that our results would not be changed if we dropped the "partial" uniformity in the de nition and simply required pointwise convergence, i.e., for all a i ; G 2 G, there exists " n! 0 s.t. h i (t i ja i ; G) < h i (t n i ja i ; G)+" n. But it is more convenient to state the upper strategic convergence property in the form of our de nition. De nition 6 (Lower Strategic Convergence) ((t n i ) 1 n=1 ; t i) satisfy the lower strategic convergence property (written t n i all n; a i ; G 2 G m.! L t i ) if 8m, 9 " n! 0 s.t. h i (t n i ja i ; G) < h i (t i ja i ; G) + " n for This implies that if a i 2 R i (t i ; G; 0), then there exists " n! 0 such that a i 2 R i (t n i ; G; " n ) for each n. This is simply the mirror image of the upper strategic convergence property and requires that h i (t n i ja i ; G) does not drop in the limit for all values of h i (t i ja i ; G) (not just for h i (t i ja i ; G) = 0). Now consider the following notion of distance between types: d (t i ; t 0 i) = X m m sup d (t i ; t 0 ija i ; G) (1) a i ;G2G m where 0 < < 1. Lemma 2 The distance d is a metric: 12 It is possibly stronger than this property since upper convergence requires that h i (t n i ja i; G) does not jump in the limit for all values of h i (t i ja i ; G) (not just for h i (t i ja i ; G) = 0). We have not veri ed if it is strictly stronger. 14

15 Proof. First note d is symmetric by de nition. To see that d satis es the triangle inequality, note that for each action a i and game G, hence d (t i ; t 00 i ja i ; G) = jh i (t i ja i ; G) h i (t 00 i ja i ; G)j d (t i ; t 00 i ) = X m jh i (t i ja i ; G) h i (t 0 ija i ; G)j + jh i (t 0 ija i ; G) h i (t 00 i ja i ; G)j = d (t i ; t 0 ija i ; G) + d (t 0 i; t 00 i ja i ; G) X m X m m m sup d (t i ; t 00 i ja i ; G) a i ;G2G m sup (d (t i ; t 0 ija i ; G) + d (t 0 i; t 00 i ja i ; G)) a i ;G2G m m sup d (t i ; t 0 ija i ; G) + sup d (t 0 i; t 00 i ja i ; G) a i ;G2G m a i ;G2G m = X sup d (t i ; t 0 ija i ; G) + X m a i ;G2G m m = d (t i ; t 0 i) + d (t 0 i; t 00 i ). m Theorem 1 below implies that d (t i ; t 0 i) = 0 ) t i = t 0 i. Lemma 3 d (t n i ; t i )! 0 if and only if t n i! U t i and t n i! L t i. Proof. Suppose d (t n i ; t i )! 0. Fix m and let Now for any a i and G 2 G m, " n = m d (t n i ; t i ). m sup d (t 0 i; t 00 i ja i ; G) a i ;G2G m so m jh i (t n i ja i ; G) h i (t i ja i ; G)j X m m sup d (t n i ; t i ja 0 i; G 0 ) = d (t n i ; t i ) ; a 0 i ;G0 2G m jh i (t n i ja i ; G) h i (t i ja i ; G)j m d (t n i ; t i ) = " n ; thus t n i! U t i and t n i! L t i. Conversely, suppose that t n i! U t i and t n i! L t i. Then 8m, 9 " n (m)! 0 and " n (m)! 0 s.t. for all a i ; G 2 G m, h i (t i ja i ; G) < h i (t n i ja i ; G) + " n (m) and h i (t n i ja i ; G) < h i (t i ja i ; G) + " n (m). 15

16 Thus d (t n i ; t i ) = X m X m m sup d (t n i ; t i ja i ; G) a i ;G2G m m max (" n (m) ; " n (m))! 0 as n! 1. De nition 7 The strategic topology is the topology generated by metric d. The strategic topology is thus a metric topology where sequences converge if and only if they satisfy upper and lower strategic convergence. In general, the closed sets in a topology are not determined by the convergent sequences, and extending the convergence de nitions to a topology can introduce more convergent sequences. However, since metric spaces are rst countable, convergence and continuity can be assessed by looking at sequences (Munkres (1975), p.190), and so this must also be the coarsest metric topology with the desired continuity properties. We now illustrate the strategic topology with the example discussed earlier. 5.1 The Example Revisited We can illustrate the de nitions in this section with the example introduced informally earlier. We show that we have convergence of types in the product topology, t 1k! t 11, corresponding to the upper-hemicontinuity noted in section 2, while d (t 1k ; t 11 ) 6! 0, corresponding to the failure of lower hemi-continuity. Writing G b for the payo s in the example, we observe that beliefs can be de ned as follows: (t 11 ) [(t 2 ; )] = (t 1m ) [(t 2 ; )] = (t 11 ) [(t 2 ; )] = ( 1, if (t2 ; ) = (t 21 ; 0) 8 >< >: 0, otherwise 1, if (t 2 2; ) = (t 2;m 1 ; 1) 1, if (t 2 2; ) = (t 2m ; 1) 0, otherwise ( 1, if (t2 ; ) = (t 21 ; 1) 0, otherwise 16, for all m = 2; 3; ::::

17 (t 21 ) [(t 1 ; )] = (t 2m ) [(t 1 ; )] = 8 >< >: 8 >< >: 1, if (t 2 1; ) = (t 11 ; 0) 1, if (t 2 2; ) = (t 12 ; 1) 0, otherwise 1, if (t 2 1; ) = (t 1m ; 1) 1, if (t 2 2; ) = (t 1;m+1 ; 1) 0, otherwise, for all m = 2; 3; :::: (t 21 ) [(t 1 ; )] = ( 1, if (t1 ; ) = (t 11 ; 1) 0, otherwise Thus we have product convergence, t 1k! t 11. Now for any " < 1+, 2 R1 0 t 1 ; G; b " t 2 ; G; b " R 0 2 = fn; Ig for all t 1 = fn; Ig for all t 2 R 1 1 R 1 2 t 1 ; G; b " t 2 ; G; b " ( fng, if t1 = t 11 = fn; Ig, if t 1 2 ft 12 ; t 13 ; :::g [ ft 11 g = fn; Ig R 2 1 R 2 2 t 1 ; G; b " t 2 ; G; b " = = ( fng, if t1 = t 11 fn; Ig, if t 1 2 ft 12 ; t 13 ; :::g [ ft 11 g ( fng, if t2 = t 21 fn; Ig, if t 2 2 ft 22 ; t 23 ; :::g [ ft 21 g R 3 1 R 3 2 t 1 ; G; b " t 2 ; G; b " = = ( fng, if t1 2 ft 11 ; t 12 g fn; Ig, if t 1 2 ft 13 ; t 14 ; :::g [ ft 11 g ( fng, if t2 = t 21 fn; Ig, if t 2 2 ft 22 ; t 23 ; :::g [ ft 21 g R1 2m t 1 ; G; b " R2 2m t 2 ; G; b " = = ( fng, if t1 2 ft 11 ; :::; t 1m g fn; Ig, if t 1 2 ft 1;m+1 ; t 1;m+2 ; :::g [ ft 11 g ( fng, if t2 2 ft 21 ; :::; t 2m g fn; Ig, if t 2 2 ft 2;m+1 ; t 2;m+2 ; :::g [ ft 21 g 17

18 for m = 2; 3; ::: for m = 2; 3; ::::; so R1 2m+1 t 1 ; G; b " R2 2m+1 t 2 ; G; b " = = ( fng, if t1 2 ft 11 ; :::; t 1;m+1 g fn; Ig, if t 1 2 ft 1;m+2 ; t 1;m+3 ; :::g [ ft 11 g ( fng, if t2 2 ft 21 ; :::; t 2m g fn; Ig, if t 2 2 ft 2;m+1 ; t 2;m+2 ; :::g [ ft 21 g R 1 t 1 ; G; b " R 2 t 2 ; G; b " = = ( fng, if t1 2 ft 11 ; t 12 ; :::g fn; Ig, if t 1 = t 11 ( fng, if t2 2 ft 21 ; t 22 ; :::g fn; Ig, if t 2 = t 21 Now observe that h 1 t 1k I; b G = 8 >< >: 2, if k = 1 1+, if k = 2; 3; ::: 2 0, if k = 1 N; (while h 1 t 1k G b = 0 for all k). Thus d (t 1k ; t 11 ) 1+ 2 have d (t 1k ; t 11 )! 0. for all k = 2; 3; :: and we do not 6 Results 6.1 The relationships among the notions of convergence We rst demonstrate that both lower strategic convergence and upper strategic convergence imply product convergence. Theorem 1 Upper strategic convergence implies product convergence. Lower strategic convergence implies product convergence. These results follow from a pair of lemmas. The product topology is generated by the metric ed(t i ; t 0 i) = X k d e k (t i ; t 0 i) k 18

19 where 0 < < 1 and e d k is a metric on the kth level beliefs that generates the topology of weak convergence. One such metric is the Prokhorov metric, which is de ned as follows. For any metric space X; let F be the Borel sets, and for A 2 F set A = fx 2 Xj inf y2a jx yj g: Then the Prokhorov distance between measures and 0 is (; 0 ) = inffj(a) 0 (A )+ for all A 2 Fg, and ~ d k (t i ; t 0 i) = ( k (t i ); k (t 0 i)): Lemma 4 For all k and c > 0, there exist " > 0 and m s.t. if e d k (t i ; t 0 i) > c, 9a i ; G 2 G m s.t. h i (t 0 ija i ; G) + " < h i (t i ja i ; G). Proof. To prove this we will construct a variant of a "report your beliefs" game, and show that any two types whose k th order beliefs di er by will lose a non-negligible amount by taking the (unique) rationalizable action of the other type. To de ne the nite games we will use for the proof, it is useful to rst think of a very large in nite action game where the action space is the type space T. Thus the rst component of player i s action is a probability distribution over : a 1 i 2 (). The second component of the action is an element of ( ()), and so on. The idea of the proof is to start with a proper scoring rule for this in nite game (so that each player has a unique rationalizable action, which is to truthfully report his type), and use it to de ne a nite game where the rationalizable actions are close to truth telling. To construct the nite game, we have agents report only the rst k levels of beliefs, and impose a nite grid on the reports at each level. Speci cally, for any xed integer z 1 let A 1 be the set of probability distributions a 1 on such that for all 2 ; a 1 () = j=z 1 for some integer j; 1 j z. Thus A 1 = fa 2 R jj : a = j=z for some integer j; 1 j z 1 ; P a = 1g; it is a discretization of the set () with grid points that are evenly spaced in the Euclidean metric. Let D 1 = A 1. Note that this is a nite set. Next pick an integer z 2 and let A 2 be the set of probability distributions on D 1 such that a 2 (d) = j=z 2 for some integer j; 1 j z 2. Continuing in this way we can de ne a sequence of nite action sets A j, where every element of each A j is a probability distribution with nite support. The overall action chosen is a vector in A 1 A 2 ::: A k. We call the a m the "m th -order action." Let the payo function be 2 X g i (a 1 ; a 2 ; ) = 2a 1 i () a 1 i ( 0 2+ kx 6 ) 42a m i a 1 j; ::; a m 1 j ; X 0 m=2 19 ea 1 j ;::;eam 1 j ; e a m i 3 ea 1 j; ::; ea m 1 j ; e

20 Note that the objective functions are strictly concave, and that the payo to the m th -order action depends only on the state and on actions of the other player up to the (m level (so the payo to a 1 i does not depend on player j s action at all). This will allow us to determine the rationalizable sets recursively, starting from the rst-order actions and working up. The rationalizable rst-order action(s) for type t i with rst-order beliefs 1 is the point or points a 1( 1 ) 2 A 1 that is closest to 1. Picking any other point involves a loss, so for each grid size z 1 there is an " > 0 such that no other actions are " rationalizable for t. Moreover, any type whose rst-order beliefs are su ciently di erent from 1 (t) will want to pick a di erent action. Thus for all c > 0, if d( e 1 (t i ); 1 (t 0 i)) > c, there are " 1 > 0 and z 1 such that h i (t 0 ija 1(t 0 i); G) + " 1 < h i (t i j 1(t 0 i); G) : This proves the claim for the case k = 1. 1) th Now let 2 (t i ) 2 ( ()) be the second-order belief of t i. For any xed rstlevel grid z 1, we know from the rst step that there is an " 1 > 0 such that for any 1; the only " 1 -rationalizable rst-order actions are the point or points a 1 in the grid that are closest to 1 : Suppose that player i believes player j is playing a rst-order action that is " 1 -rationalizable. Then player i s beliefs about the nite set D 1 = A 1 correspond to the nite-dimensional measure 2, where the probability of measurable X A 1 is 2 (X) = 2 (f( 1 ; ) : (; a 1( 1 )) 2 Xg). That is, for each 1 that i thinks j could have, i expects that j will play an element of the corresponding a 1( 1 ). Because A 2 is a discretization of (D 1 ); player i may not be able to chose a 2 = 2. However, because of the concavity of the objective function, the constrained second-order best reply of i with beliefs 2 is the point a 2 2 A 2 that is closest to 2 in the Euclidean metric, and choosing any other action incurrs a non-zero loss. Moreover, a 2 is at (Euclidean) distance from 2 that is bounded by the distance between grid points, so there is a bound on the distance that goes to zero as z 2 goes to in nity, uniformly over all 2. Next we claim that if there is a c > 0 such that d 2 (t i ; t 0 i) > c; then 2(t i ) 6= 2(t 0 i) for all su ciently ne grids A 1 in ( 1 ). To see this, note from the de nition of the Prokhorov metric, if d e2 (t i ; t 0 i) > c there is a Borel set A in () such that 2 (t i )(A) > 2 (t 0 i)(a c )+c. Because the rst-order actions a 1 converge uniformly to 1 as z 2 goes to in nity, (; a 1 ( 1 )) 2 A c for every (; 1 ) 2 A, so for all such that c=2 > > 0 there is a z 2 such that 2(t i )(A c ) 20

21 2(t i )(A) > 2 (t i )(A) > 2 (t 0 i)(a c ) +c 2(t 0 i) (A c ) 2+c > 2(t 0 i) (A c ), where the rst inequality follows from set inclusion, the second and fourth from the uniform convergence of the a 1, and the third from d e2 (t i ; t 0 i) > c. From the previous step, this implies that when d 2 (t i ; t 0 i) > c there is a z 2 and " 2 > 0 such that h i (t 0 ija 2(t 0 i); G) + " 2 < h i (t i j 2(t 0 i); G) for all z 2 > z 2. We can continue in this way to prove the result for any k. Lemma 5 Suppose that t i is not the limit of the sequence t n i in the product topology, then (t n i ; t i ) satis es neither the lower convergence property nor the upper convergence property. Proof. Failure of product convergence implies that there exists k such that e d k (t n i ; t i ) does not converge to zero, so there exists > 0 such that (for some subsequence) ed k (t n i ; t i ) > for all n. By lemma 4, there exists " and m such that, for all n, (a) 9a i ; G 2 G m s.t. h i (t i ja i ; G) + " < h i (t n i ja i ; G) and (b) 9a i ; G 2 G m s.t. h i (t n i ja i ; G) + " < h i (t i ja i ; G) : Now suppose that the lower convergence property holds. Then there exists n! 0 such that h i (t n i ja i ; G) < h i (t i ja i ; G) + n for all a i ; G 2 G m. This combined with (a) gives a contradiction. Similarly, upper convergence implies that there exists n! 0 such that h i (t i ja i ; G) < h i (t n i ja i ; G) + n for all a i ; G 2 G m. This give a contradiction when combined with (b). Lemma 5 immediately implies theorem 1. Theorem 2 Product convergence implies upper strategic convergence. 21

22 Proof. Suppose that t n i product-converges to t i. If upper strategic convergence fails there is an m; a i ; G 2 G m s.t. for all " n! 0 and N, there is n 0 > N such that h i (t i ja i ; G) > h i t n0 i ja i ; G + " n0. We may relabel so that t n i is the subsequence where this inequality holds. Pick so that h i (t i ja i ; G) > h i (t n i ja i ; G) + for all n. Since, for each n and t n i, a i 2 R i (t n i ; G; h i (t i ja i ; G) ), there exists n 2 Tj A j s.t. (i) n [f(t j ; ; a j ) : a j 2 R j (t j ; G; h i (t i ja i ; G) ) for all j 6= ig] = 1 (ii) marg T n = (t n i ) " # R gi (a i ; a j ; ) (iii) d n g i (a 0 i; a j ; ) (t j ;;a j ) h i (t i ja i ; G) + for all a 0 i 2 A i Since under the product topology, T is a compact metric space, and since A j and are nite, so is T A j. Thus (T A j ) is compact in the weak topology, so the sequence n has a limit point,. Now since (i), (ii) and (iii) holds for every n and = lim n n, we have (i*) [f(t j ; ; a j ) : a j 2 R j (t j ; G; h i (t i ja i ; G) )g] = 1 (ii*) marg T = (t i ) " # R gi (a i ; a j ; ) (iii*) d g i (a 0 i; a j ; ) (t j ;;a j ) h i (t i ja i ; G) + for all a 0 i 2 A i where (i*) follows from the fact that f(t j ; ; a j ) : a j 2 R j (t j ; G; h i (t i ja i ; G) This implies a i 2 R i (t i ; G; h i (t i ja i ; G) ), a contradiction. )g is closed. Corollary 2 Lower strategic convergence implies convergence in the strategic topology. Proof. We have t n i! L t i ) t n i! t i (by theorem 1); t n i! t i ) t n i! U t i (by theorem 2); and t n i! L t i and t n i! U t i ) d (t n i ; t i )! 0 (by lemma 3). 22

23 6.2 Finite types are dense in the strategic topology Theorem 3 Finite types are dense under d. Given corollary 2, the theorem follows from lemma 6 below, which shows that, for any type in the universal type space, it is possible to construct a sequence of nite types that lower converge to it. The proof of lemma 6 is long, and broken into many steps. In outline, we rst nd a nite grid of games that approximate all games with m actions. We then map types onto their "epsilons" for action/game pairs from this nite set, impose a grid on these epsilons, and de ne a belief hierarchy for a nite type by arbitrary taking the belief hierarchy of one of the types that is mapped to it. This gives us a nite type space. We show that this map "preserve the epsilons": a i 2 R i (t i ; g; ") ) a i 2 R i (f i (t i ) ; g; "). Finally we show that for any type in the universal space there is a sequence of these nite types that "lower converge" to it. Our proof thus follows Monderer and Samet (1996) in constructing a mapping from types in one type space to types in another type space that preserves approximate best response properties. Their construction worked for equilibrium, while our construction works for rationalizability, and thus the approximation has to work for many conjectures over opponents play simultaneously. We assume neither a common prior nor a countable number of types, and we develop a topology on types based on the play of the given types as opposed to a topology on priors or information systems. 13 A distinctive feature of our work is that we identify types in our constructed type space with sets of "-rationalizable actions for a nite set of " and a nite set of games. The recent paper of Ely and Peski (2005) similarly identi es types with sets of rationalizable actions, although for their di erent purpose (constructing a universal type space for the independent interim rationalizability solution concept), no approximation is required. Lemma 6 For any t i, there exists a sequence of nite types t n i lower strategic convergence. such that [(t n i ) 1 n=1 ; t i] satisfy For the rst part of the proof, we will restrict attention to m action games with A 1 = A 2 = f1; 2; :; mg. Having xed the action sets, a game is parameterized by the payo 13 It is not clear how one could develop a topology based on the equilibrium distribution of play in a setting without a common prior. At a technical level, the fact that the strategic topology is not uniform over the number of actions requires extra steps to deal with the cardinality of the number of actions. (See Section 7.2 for a brief discussion of uniformity.) 23

24 function g : A 1 A 2! [ M; M] 2. For a xed m, we will write g for the game G = (f1; 2; :; mg ; f1; 2; :; mg ; g). Let D (g; g 0 ) = sup jg i (a; ) gi 0 (a; )j. i;a; Some preliminary lemmas are proved in the Appendix. There is a nite grid of games with m actions such all games are within " of a game in the grid: Lemma 7 For any integer m and " > 0, there exists a nite collection of m action games G such that, for every g 2 G m, there exists g 0 2 G such that D (g; g 0 ) ". The rationalizability of an action (h i (t i ja i ; g)) is close to the rationalizability of that action in nearby games: Lemma 8 For all i, t i, a i, g and g 0, for all i, a i. h i (t i ja i ; g) h i (t i ja i ; g 0 ) + 2D (g; g 0 ) So the rationalizability of an action is close to the rationalizability of that action in a nearby game in the grid: Lemma 9 For any integer m and " > 0, there exists a nite collection of m action games G such that, for every g 2 G m, there exists g 0 2 G such that for all i, t i, a i, g and g 0. jh i (t i ja i ; g) h i (t i ja i ; g 0 )j ". The main step of the proof of lemma 6 is then the construction of a nite type space that replicates the "-rationalizability properties all games in the grid of m action games if " is in some nite grid. Lemma 10 Fix any nite collection of m action games G and > 0. There exists a nite type space (T i ; b i ) i=1;2 and functions (f i ) i=1;2, each f i : T! T i, such that R i (t i ; g; ") R i (f i (t i ) ; g; ") for all t i 2 T and " 2 f0; ; 2; ::::g. 24

25 Proof. Write hxi for the smallest number in the set = f0; ; 2; g greater than x. Let be de ned by n f i : T! b t i : G A! n0; ; ::::; h2mi oo f i (t i ) [g; a i ] = hh i (t i ja i ; g)i n n for all a i and g 2 G. Let T i b t i : G A! 0; ; ::::; h2mi oo be the range of f i. Note that T i is nite set by construction. De ne b i : T i! (T j ) as follows. For each bt i 2 T i, x any t i 2 T such that f i (t i ) = bt i. Label this type i bt i and let Fix " 2 b i bt i b t j ; = i i bt i (tj ; ) : f j (t j ) = bt j. n 0; ; ::::; h2mi o. Let S i bt i = Ri i bt i ; g; ". We argue that S is an "-best response set on the type space (T i ; b i ) i=1;2. To see why, observe that a i 2 R i i bt i ; g; " implies that there exists 2 (T A j ) such that [f(t j ; ; a j ) : a j 2 R j (t j ; g; ")g] = 1 marg T Z " # gi (a i ; a j ; ) = i i bt i (t j ;;a j ) g i (a 0 i; a j ; ) d " for all a 0 i 2 A i Now de ne b 2 (T j A j ) by b bt j ; ; a j = (tj ; ; a j ) : f j (t j ) = bt j By construction, b bt j ; ; a j : aj 2 S j bt j = 1 marg Tj b = b i bt i " # X gi (a i ; a j ; ) b bt j ; ; a j " for all a 0 g i (a 0 i 2 A i i; a j ; ) a i ;a j ;bt j ; 25

26 So a i 2 BR i (S) (t i ). Since S is an "-best response set on the type space (T i ; b i ) i=1;2, S i bt i Ri bt i ; g; ". Thus a i 2 R i (t i ; g; ") ) a i 2 R i (f i (t i ) ; g; "). Now the same nite type space constructed in the proof of lemma 10 can then be shown to approximately replicate the "-rationalizability properties of all games in the grid of m action games for all ": Lemma 11 Fix any nite collection of m action games G and > 0. There exists a nite type space (T i ; b i ) i=1;2 and functions (f i ) i=1;2, each f i : T! T i, such that h i (f i (t i ) ja i ; g) h i (t i ja i ; g) + for all t i, g 2 G and a i. And the same nite type space approximately replicates the "-rationalizability properties of all m action games: Lemma 12 Fix the number of actions m and > 0. There exists a nite type space (T i ; b i ) i=1;2 and functions (f i ) i=1;2, each f i : T! T i, such that h i (f i (t i ) ja i ; g) h i (t i ja i ; g)+ for all t i, g 2 G m and a i. Now the proof of lemma 6 can be completed as follows. Lemma 12 implies that for any integer m, there exists a nite type bt m i such that h i bt m i ja i ; G h i (t i ja i ; G) + 1 m for all a i and G 2 G m. Now x any m and let ( 2M, if n m " n = 1, if n > m. n Observe that " n! 0, h i bt n i ja i ; G h i (t i ja i ; G) + 2M = h i (t i ja i ; G) + " n for all a i and G 2 G m if n m and h i bt n i ja i ; G h i (t i ja i ; G) + 1 n = h i (t i ja i ; G) + " n for all a i and G 2 G m if n > m. 26

27 7 Discussion The key implication of our denseness result is that there are "enough" nite types to approximate general ones. If there is some strategic behavior that arises in some game on the universal type space, then there exists a nite type for which that behavior arises. In this section we discuss some caveats regarding the interpretation of this result. First, we show that there is a sense in which the set of nite types is small. Then we discuss a topology that is uniform over all games. The denseness result does not hold with such a topology, and hence the same nite type cannot approximate strategic behavior for an in nite type in all games simultaneously. However, the approximation does hold for all games with a bounded number of actions. Moreover, we believe that in cases where arbitrarily large action spaces are of interest, there is usually a natural metric on actions that makes nearby actions similar. Then we discuss relaxing the uniform bound on payo s that we have used throughout the paper. Last, we emphasize the caution needed in working with nite types despite our result. 7.1 Is the set of nite types "generic"? Our denseness result does not imply that the set of nite types is "generic" in the universal type space. While it is not obvious why this question is important from a strategic point of view, we nonetheless brie y report some results showing that the set of nite types is not generic in either of two standard topological senses. First, a set is sometimes said to be generic if it is open and dense. But the set of nite types is not open. To show this, it is enough to show that the set of in nite types is dense. This implies that the set of in nite types is not closed and so the set of nite types is not open. We write Tn be the collection of all types that exist on nite belief closed subsets of the universal type space where each player has at most n types. The set of nite types is the countable union T F = [ n Tn. The set of in nite types is the complement of T F in T. 27